[Paper Review] Thermostatistics with minimal length uncertainty relation

Babak Vakili, Mohammad Ali Gorji|arXiv (Cornell University)|Jul 4, 2012

Statistical Mechanics and Entropy3 citations

TL;DR

This paper develops statistical mechanics within the Generalized Uncertainty Principle (GUP) framework, which incorporates a minimal length scale from quantum gravity. By deforming phase space structure via a generalized commutation relation, the authors derive modified Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics, revealing a finite upper bound on excited particles at high temperatures—indicating a novel quantum gravity-induced condensation not present in classical statistics.

ABSTRACT

Existence of minimal length is suggested in any quantum theory of gravity such as string theory, double special relativity and black hole physics. One way to impose minimal length is deforming Heisenberg algebra in phase space which is called Generalized Uncertainty Principle (GUP). In this paper, we develop statistical mechanics in GUP framework. Our method is quite general and does not need to fix the generalized coordinates and momenta. We define general transformation in phase space which transforms usual Heisenberg algebra to a deformed one. In this method, quantum gravity effects only acts on the structure of phase space and we relate these effects to the density of states. We find an interesting phenomenon in Maxwell-Boltzmann statistics which has not a classical analogy. We show that there is an upper bound for the number of excited particles in the limit of high temperature which implies to the condensation. Also we study modification of Bose-Einstein condensation and the completely degenerate gas.

Motivation & Objective

• To extend statistical mechanics to include quantum gravity effects via the Generalized Uncertainty Principle (GUP), which introduces a minimal length scale.
• To develop a general framework for thermostatistics without fixing specific coordinates or momenta, focusing on phase space deformation.
• To investigate how quantum gravity corrections, encoded in the GUP, modify the density of states and statistical distributions.
• To analyze the thermodynamic behavior of ideal gases, especially Bose-Einstein condensation and completely degenerate gases, under GUP corrections.
• To identify novel quantum gravity signatures in statistical systems, such as an upper bound on excited particles at high temperatures.

Proposed method

• Derives a deformed Heisenberg algebra via the GUP, introducing a minimal length through the commutation relation $[\hat{X}_i, \hat{P}_j] = i\hbar(1 + \beta \mathbf{P}^2)\delta_{ij}$.
• Introduces a general phase space transformation that maps standard Heisenberg algebra to the deformed GUP algebra, preserving generality.
• Modifies the density of states in phase space up to second order in $\beta$, using $a(\varepsilon)d\varepsilon \simeq \frac{d^3Xd^3P}{h^3}(1 - 3\beta P^2 + 6\beta^2 P^4 + \cdots)$.
• Applies the modified density of states to derive particle density and pressure in terms of generalized Fermi-Dirac and Bose-Einstein functions.
• Expands the particle density and pressure expressions in powers of $\beta$ for general power-law energy-momentum relations $\varepsilon = \eta P^\alpha$.
• Uses the function $h_\nu(z)$ to express integrals in terms of standard Fermi-Dirac and Bose-Einstein functions, enabling analytical treatment of corrections.

Experimental results

Research questions

• RQ1 How does the existence of a minimal length, as implied by quantum gravity, modify the standard statistical mechanics framework?
• RQ2 What are the effects of GUP-deformed phase space structure on the density of states and statistical distributions in many-body systems?
• RQ3 Does the GUP framework lead to new thermodynamic behaviors, such as a finite upper bound on excited particles at high temperatures?
• RQ4 How are Bose-Einstein condensation and degenerate quantum gases modified under the GUP compared to standard quantum statistics?
• RQ5 Can the modified particle density and pressure be expressed analytically in terms of standard special functions with GUP corrections?

Key findings

• The GUP framework introduces a finite upper bound on the number of excited particles in Maxwell-Boltzmann statistics at high temperatures, implying a novel form of condensation not present in classical statistics.
• The modified particle density includes corrections proportional to $\beta (T/\eta)^{2/\alpha}$ and $\beta^2 (T/\eta)^{4/\alpha}$, which depend on the energy-momentum relation $\varepsilon = \eta P^\alpha$.
• For the ultra-relativistic case ($\alpha = 1$), the particle density correction scales as $\beta T^2$, showing a significant deviation from standard statistics at high temperatures.
• The pressure is also modified through higher-order corrections in $\beta$, with the leading correction term involving $\beta (T/\eta)^{2/\alpha}$ and the ratio of Gamma and $h_\nu(z)$ functions.
• The modified Bose-Einstein and Fermi-Dirac distributions retain their functional form but are multiplied by a series of $\beta$-dependent corrections, indicating a deformation of the underlying phase space structure.
• The results suggest that quantum gravity effects, encoded in the GUP, can lead to observable deviations in thermodynamic quantities, particularly in high-energy or high-temperature regimes.